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Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded ..."
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic
On Miura Transformations and VolterraType Equations Associated with the Adler–Bobenko–Suris Equations
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterratype equations. We show that the ABS equations correspond to Bäcklund ..."
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Cited by 15 (8 self)
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We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterratype equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever–Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.
1 MATRIX ALPS: Accelerated Low Rank and Sparse Matrix Reconstruction
"... We propose MATRIX ALPS for recovering a sparse plus lowrank decomposition of a matrix given its corrupted and incomplete linear measurements. Our approach is a firstorder projected gradient method over nonconvex sets, and it exploits a wellknown memorybased acceleration technique. We theoretica ..."
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We propose MATRIX ALPS for recovering a sparse plus lowrank decomposition of a matrix given its corrupted and incomplete linear measurements. Our approach is a firstorder projected gradient method over nonconvex sets, and it exploits a wellknown memorybased acceleration technique. We theoretically characterize the convergence properties of MATRIX ALPS using the stable embedding properties of the linear measurement operator. We then numerically illustrate that our algorithm outperforms the existing convex as well as nonconvex stateoftheart algorithms in computational efficiency without sacrificing stability. I.
Cultural Diversity, Geographical Isolation, and the Origin of the Wealth of Nations
, 2012
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Sequences of periodic solutions and infinitely many coexisting attractors in the bordercollision normal form
, 2013
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Mortality of Iterated Piecewise Affine Functions over the Integers: Decidability and Complexity
, 2013
"... In the theory of discretetime dynamical systems one studies the limiting behaviour of processes defined by iterating a fixed function f over a given space. A muchstudied case involves piecewise affine functions on Rn. Blondel et al. (2001) studied the decidability of questions such as global conve ..."
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In the theory of discretetime dynamical systems one studies the limiting behaviour of processes defined by iterating a fixed function f over a given space. A muchstudied case involves piecewise affine functions on Rn. Blondel et al. (2001) studied the decidability of questions such as global convergence and mortality for such functions with rational coefficients. Mortality means that every trajectory includes a 0; if the iteration is implemented as a loop while (x = 0) x: = f(x), mortality means that the loop is guaranteed to terminate. Checking the termination of simple loops (under various restrictions of the guard and the update function) is a muchstudied topic in automated program analysis. Blondel et al. proved that the problems are undecidable when the state space is R n (or Q n), and the dimension n is at least two. From a program analysis (and discrete Computability) viewpoint, it is more natural to consider functions over the integers. This paper establishes (un)decidability results for the integer setting. We show that also over integers, undecidability (moreover, Π 0 2 completeness) begins at two dimensions. We further investigate the effect of several restrictions on the iterated functions. Specifically, we consider bounding the size of the partition defining f, and restricting the coefficients of the linear components. In the decidable cases, we give complexity results. The complexity is PTIME for affine functions, but for piecewiseaffine ones it is PSPACEcomplete. The undecidability proofs use some variants of the Collatz problem, which may be of independent interest. 1
Dynamic Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded ..."
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic dynamics with hybrid systems. We survey dynamic logics for specifying and verifying properties for each of those classes of dynamical systems. A dynamic logic is a firstorder modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of firstorder modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about
Resolving Gödel's incompleteness myth: Polynomial Equations and Dynamical Systems for Algebraic Logic, arXiv:1112.2141 [math.GM
, 2011
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Towards a Unified Theory of Economic Growth: Oded Galor on the Transition from Malthusian Stagnation to Modern Economic Growth By
"... An interview with Oded Galor on the development of unified growth theory. ..."
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An interview with Oded Galor on the development of unified growth theory.
Active Queue Management via EventDriven Feedback Control
"... Active Queue Management (AQM) is investigated to avoid incipient congestion in gateways to complement congestion control run by the transport layer protocol such as the TCP. Most existing work on AQM can be categorized as (1) adhoc eventdriven control and (2) timedriven feedback control approache ..."
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Active Queue Management (AQM) is investigated to avoid incipient congestion in gateways to complement congestion control run by the transport layer protocol such as the TCP. Most existing work on AQM can be categorized as (1) adhoc eventdriven control and (2) timedriven feedback control approaches based on control theory. Ad hoc eventdriven approaches for congestion control, such as RED (Random Early Detection), lack a mathematical model. Thus, it is hard to analyze their dynamics and tune the parameters. Timedriven control theoretic approaches based on solid mathematical models have drawbacks too. As they sample the queue length and run AQM algorithm at every fixed time interval, they may not be adaptive enough to an abrupt load surge. Further, they can be executed unnecessarily often under light loads due to the timedriven nature. To seamlessly integrate the advantages of both eventdriven and controltheoretic timedriven approaches, we present an eventdriven feedback control approach based on formal control theory. As our approach is based on a mathematical model, its performance is more analyzable and predictable than ad hoc eventdriven approaches are. Also, it is more reactive to dynamic load changes due to its eventdriven nature. Our simulation results show that our eventdriven controller effectively maintains the queue length around the specified setpoint. It achieves shorter E2E (endtoend) delays and smaller E2E delay fluctuations than several existing AQM approaches, which are adhoc eventdriven